Roman numerals

Roman numerals are a numeral system originating in ancient Rome, adapted from Etruscan numerals. The system used in classical antiquity was slightly modified in the Middle Ages to produce a system used today. It is based on certain letters which are given values as numerals.

Roman numerals are commonly used in numbered lists (in outline format), clock faces, pages preceding the main body of a book, chord triads in music analysis, the numbering of movie publication dates, successive political leaders or children with identical names, and the numbering of some annual sport events. See Modern usage below.

Symbols

Multiple symbols may be combined to produce numbers in between these values, subject to certain rules on repetition. In cases where it may be shorter, it is sometimes allowable to place a smaller, subtractive, symbol before a larger value, so that, for example, one may write IV or iv for four, rather than IIII. Sometimes, especially in medical prescriptions, a final i becomes j, such as iij for 3 or vij for 7. This stems from the medieval use of j as a swash character (a form of i) to end numbers. Again, for the numbers not assigned a specific symbol, the above given symbols are combined:

For large numbers (4000 and above), a bar can be placed above a base numeral to indicate multiplication by 1000, although the Romans themselves often just wrote out the "M"s:[1]

V for five thousand

X for ten thousand

L for fifty thousand

C for one hundred thousand

D for five hundred thousand

M for one million

The basic Roman numerals follow a pattern:

Units

I

II

III

IV

V

VI

VII

VIII

IX

Tens

X

XX

XXX

XL

L

LX

LXX

LXXX

XC

Hundreds

C

CC

CCC

CD

D

DC

DCC

DCCC

CM

Thousands

M

MM

MMM

IV

V

VI

VII

VIII

IX

Ten Thousands

X

XX

XXX

XL

L

LX

LXX

LXXX

XC

100 Thousands

C

CC

CCC

CD

D

DC

DCC

DCCC

CM

Origins

Although the Roman numerals are now written with letters of the Roman alphabet, they were originally independent symbols. The Etruscans, for example, used I Λ X ⋔ 8 ⊕ for I V X L C M, of which only I and X happened to be letters in their alphabet. One folk etymology has it that the V represented a hand, and that the X was made by placing two Vs on top of each other, one inverted. However, the Etrusco-Roman numerals actually appear to derive from notches on tally sticks, which continued to be used by Italian and Dalmatian shepherds into the 19th century.[2]

Thus I descends not from the letter I but from a notch scored across the stick. Every fifth notch was double cut (i.e. ⋀, ⋁, ⋋, ⋌, etc.), and every tenth was cross cut (X), IIIIΛIIIIXIIIIΛIIIIXII…, much like European tally marks today. This produced a positional system: Eight on a counting stick was eight tallies, IIIIΛIII, or the eighth of a longer series of tallies; either way, it could be abbreviated ΛIII (or VIII), as the existence of a Λ implies four prior notches. By extension, eighteen was the eighth tally after the first ten, which could be abbreviated X, and so was XΛIII. Likewise, number four on the stick was the I-notch that could be felt just before the cut of the Λ (V), so it could be written as either IIII or IΛ (IV). Thus the system was neither additive nor subtractive in its conception, but ordinal. When the tallies were transferred to writing, the marks were easily identified with the existing Roman letters I, V, X.

The tenth V or X along the stick received an extra stroke. Thus 50 was written variously as N, И, K, Ψ, ⋔, etc., but perhaps most often as a chicken-track shape like a superimposed V and I - ᗐ. This had flattened to ⊥ (an inverted T) by the time of Augustus, and soon thereafter became identified with the graphically similar letter L. Likewise, 100 was variously Ж, ⋉, ⋈, H, or as any of the symbols for 50 above plus an extra stroke. The form Ж (that is, a superimposed X and I) came to predominate. It was written variously as >I< or ƆIC, was then abbreviated to Ɔ or C, with C variant finally winning out because, as a letter, it stood for centum, Latin for "hundred".

The hundredth V or X was marked with a box or circle. Thus 500 was like a Ɔ superposed on a ⋌ or ⊢ — that is, like a Þ with a cross bar,— becoming D or Ð by the time of Augustus, under the graphic influence of the letter D. It was later identified as the letter D, perhaps as an abbreviation of demi-mille "half-thousand"; this at least was the folk etymology given to it later on.

Meanwhile, 1000 was a circled or boxed X: Ⓧ, ⊗, ⊕, and by Augustinian times was partially identified with the Greek letter Φ phi. In different traditions it then evolved along several different routes. Some variants, such as Ψ and CD (the latter more accurately a mirror image of a D, which is not supported by Unicode, adjacent to a regular D), were historical dead ends, although folk etymology later identified D for 500 as graphically half of Φ for 1000 because of the CD variant. A third line, ↀ, survives to this day in two variants:

One, CIƆ, led to the convention of using parentheses to indicate multiplication by a thousand: the original CIƆ = (I) 1000, then (III) for 3000, (V) 5000, (IX) 9000, (X) 10 000, (L) 50 000, (C) 100 000, (D) 500 000, (M) 1000 000, etc. This was later extended to double parentheses, as in ↁ , ↂ, etc. See #Alternate forms below.

In the other, ↀ became ∞ and ⋈, eventually changing to M under the influence of the Latin word mille "thousand".

Zero

In general, the number zero did not have its own Roman numeral, but a primitive form (nulla) was known by medieval computists (responsible for calculating the date of Easter). They included zero (via the Latin word nulla meaning "none") as one of nineteen epacts, or the age of the moon on March 22. The first three epacts were nullae, xi, and xxii (written in minuscule or lower case). The first known computist to use zero was Dionysius Exiguus in 525. Only one instance of a Roman numeral for zero is known. About 725, Bede or one of his colleagues used the letter N, the initial of nullae, in a table of epacts, all written in Roman numerals.

Fractions

A triens coin (1/3 or 4/12 of an as). Note the four dots •••• indicating its value.

Though the Romans used a decimal system for whole numbers, reflecting how they counted in Latin, they used a duodecimal system for fractions, because the divisibility of twelve (12 = 3 × 2 × 2) makes it easier to handle the common fractions of 1/3 and 1/4 than does a system based on ten (10 = 2 × 5). On coins, many of which had values that were duodecimal fractions of the unit as, they used a tally-like notational system based on twelfths and halves. A dot • indicated an uncia "twelfth", the source of the English words inch and ounce; dots were repeated for fractions up to five twelfths. Six twelfths (one half) was abbreviated as the letter S for semis "half". Uncia dots were added to S for fractions from seven to eleven twelfths, just as tallies were added to V for whole numbers from six to nine.

Each of these fractions had a name, which was also the name of the corresponding coin:

The arrangement of the dots was variable and not necessarily linear. Five dots arranged like :·: (as on the face of a die) are known as a quincunx from the name of the Roman fraction/coin. The Latin words sextans and quadrans are the source of the English words sextant and quadrant.

Other Roman fractions include:

1/8 sescuncia, sescunciae (from sesqui- + uncia, i.e. 1½ uncias), represented by a sequence of the symbols for the semuncia and the uncia.

IIII vs. IV

The notation of Roman numerals has varied through the centuries. Originally, it was common to use IIII to represent four, because IV represented the Roman godJupiter, whose Latin name, IVPPITER, begins with IV. The subtractive notation (which uses IV instead of IIII) has become universally used only in modern times. For example, Forme of Cury, a manuscript from 1390, uses IX for nine, but IIII for four. Another document in the same manuscript, from 1381, uses IV and IX. A third document in the same manuscript uses IIII, IV, and IX. Constructions such as IIIII for five, IIX for eight or VV for 10 have also been discovered. Subtractive notation arose from regular Latin usage: the number 18 was duodeviginti or “two from twenty”; the number 19 was undeviginti or "one from twenty". The use of subtractive notation increased the complexity of performing Roman arithmetic, without conveying the benefits of a full positional notation system.

An inscription on Admiralty Arch, London. The numeral translates to 1910.

Likewise, on some buildings it is possible to see MDCCCCX, for example, representing 1910 instead of MCMX – notably Admiralty Arch in London. The Leader Building in Cleveland, Ohio, at the corner of Superior Avenue and E.6th Street, is marked MDCCCCXII, representing 1912. Another notable example is on Harvard Medical School's Gordon Hall, which reads MDCCCCIIII for 1904. In Dubrovnik, Croatia, a commemorative inscription marking the 1000th anniversary of King Tomislav’s coronation (Croatia’s first King), appears as DCCCCXXV - MDCCCCXXV (925 -1925).

Calendars and clocks

The Shepherd gate clock with Roman numbers up to XXIII (and 0), in Greenwich

Clock faces that are labeled using Roman numerals conventionally show IIII for four o'clock and IX for nine o'clock, using the subtractive principle in one case and not the other. There are many suggested explanations for this, several of which may be true:

Louis XIV, king of France, who preferred IIII over IV, ordered his clockmakers to produce clocks with IIII and not IV, and thus it has remained.[3]

Using the standard numerals, two sets of figures would be similar and therefore confuseable by children and others unused to reading clockfaces: IV and the VI; and IX and XI. Since the first pair are additionally upside down on the face, an added level of confuseability would be introduced. Better, then, to make greater character distinction between them by using IIII and VI

The four-character form IIII creates a visual symmetry with the VIII on the other side, which the character IV would not.

With IIII, the number of symbols on the clock totals twenty I's, four V's, and four X's, so clock makers need only a single mold with a V, five I's, and an X in order to make the correct number of numerals for their clocks: VIIIIIX. This is cast four times for each clock and the twelve required numerals are separated:

V IIII IX

VI II IIX

VII III X

VIII I IX

The IIX and one of the IX’s are rotated 180° to form XI and XII. The alternative with IV uses seventeen I's, five V's, and four X's, requiring the clock maker to have several different molds.

Only the I symbol would be seen in the first four hours of the clock, the V symbol would only appear in the next four hours, and the X symbol only in the last four hours. This would add to the clock's radial symmetry.

Chemistry

As it relates to the nomenclature of inorganic compounds. For example MnO2 should be named manganese(IV) oxide; manganese(IIII) oxide is not used.

XCIX vs. IC

Rules regarding Roman numerals often state that a symbol representing 10n may not precede any symbol larger than 10n+1. For example, C cannot be preceded by I or V, only by X (or, of course, by a symbol representing a value equal to or larger than C). Thus, one should represent the number ninety-nine as XCIX (using decimal places -- 90 (XC) then 9 (IX)), not as the "shortcut" IC.

This problem manifested in such questions as why 1990 was not written as MXM instead of the universal usage MCMXC, or why 1999 was not written simply IMM or MIM as opposed to the universal MCMXCIX.

However, these rules are not universally followed.

Modern Roman numerals (i.e. under the rule above) are written by expressing each digit separately starting with the left most digit and skipping any digit with a value of zero. To see this in practice, consider the above example of 1990. In Roman numerals 1990 is rendered: 1000=M, 900=CM, 90=XC; resulting in MCMXC. 2008 is written as 2000=MM, 8=VIII; or MMVIII.

Year in Roman numerals

In seventeenth-century Europe, using Roman numerals for the year of publication for books was standard; there were many other places it was used as well. Publishers attempted to make the number easier to read by those more accustomed to Arabic positional numerals. On British title pages, there were often spaces between the groups of digits: M DCC LX I (relating to 1000 700 60 1 or 1761) is one example. This may have come from the French, who separated the groups of digits with periods, as: M.DCC.LXI. or M. DCC. LXI. Notice the period at the end of the sequence; many countries did this for Roman numerals in general, but not necessarily Britain. (Periods were also common on each side of numerals in running text, as in "commonet .iij. viros illos".)

These practices faded from general use before the start of the twentieth century, though the cornerstones of major buildings still occasionally use them. Roman numerals are today still used on building faces for dates: 2008 can be represented as MMVIII. They are also sometimes used in the credits of movies and television programs to denote the year of production, particularly programs made by the BBC and CBS.

Modern usage

Roman numerals remained in common use until about the 14th century, when they were replaced by Arabic numerals (thought to have been introduced to Europe from al-Andalus, by way of Arab traders and arithmetic treatises, around the 11th century). The Roman number system is generally regarded as obsolete in modern usage, but is still seen occasionally. Classical numbering is often used to suggest importance or timelessness, or in other cases where an alternate numbering system is useful for clarity. Examples of their current use include:

Names of monarchs and Popes are still displayed in Roman numerals, e.g. Elizabeth II, Benedict XVI. These are ordinal numbers; e.g. "II" is pronounced "the second".

The year and, if any, credits shown at the end of a television show or film, especially in the United Kingdom.

Some faces of clocks and timepieces show hours in Roman numerals.

Books (particularly older ones) are dated in Roman numerals, and display preliminary pages in Roman numbers. Volume numbers on spines and chapter numbers can also be in Roman numerals.

Film series and sequels of novels and video games are often numbered with Roman numerals, typically done in emulation of older books.

Outlines use I, II, III and i, ii, iii as part of their organizational structure.

Sometimes the numerals are written using lower-case letters (thus: i, ii, iii, iv, etc.), particularly if numbering paragraphs or sections within chapters, or for the pagination of the front matter of a book.

In chemistry, Roman numerals were formerly used to denote the group in the periodic table of the elements. But there was not international agreement as to whether the group of metals which dissolve in water should be called Group IA or IB, for example, so although references may use them, the international norm has recently switched to Arabic numerals. However, Roman numerals are still used in the IUPAC nomenclature of inorganic chemistry, for the oxidation number of cations which can take on several different positive charges. For example, FeO is iron(II) oxide and Fe2O3 is iron(III) oxide. In contrast, Arabic numerals are used to denote the formal oxidation state (which is not always the same as the oxidation number) of positively or negatively charged atoms. They are also used for naming phases of polymorphic crystals, such as ice.

In astronomy, the natural satellites or "moons" of the planets are traditionally designated by capital Roman numerals, at first by order from the center of the planet, as the four Galilean satellites of Jupiter are numbered, and later by order of discovery; e.g., Callisto was "Jupiter IV" or "J IV". This is particularly amusing in the case of Callisto, because, as mentioned above, the notation IV was mostly disused by the Romans for its similarity to the first two letters of Jupiter. With recent discoveries—Jupiter currently has 63 known satellites—as well as computerization, this is somewhat disparaged for the minor worlds, at least in computerized listings.

Science fiction, and not astronomy per se, has adopted the use for numbering the planets around a star; e.g., Planet Earth is called "Sol III".

In photography, Roman numerals (with zero) are used to denote varying levels of brightness when using the Zone system.

In earthquakeseismology, Roman numerals are used to designate degrees of the Mercalli intensity scale.

Music theory

In music theory, while scale degrees are typically represented with Arabic numerals, often modified with a caret or circumflex, the triads that have these degrees as their roots are often identified by Roman numerals (as in chord symbols). See also diatonic functions. Upper-case Roman numerals indicate major triads while lower-case Roman numerals indicate minor triads, as the following chart illustrates. Lower-case Roman numerals with a degree symbol indicate diminished triads. For example, in the major mode the triad on the seventh scale degree, the leading tone triad is diminished.

Also in music theory, individual strings of stringed instruments, such as the violin, are often denoted by Roman numerals, with higher numbers denoting lower strings. For example I signifies the E string on the violin and the A string on the viola and cello, these being the highest strings, respectively, on each instrument. They are also sometimes used to signify position. In this case, the number in Roman numerals corresponds with the position number. For example, III means third position and V means fifth.

Roman numeral

I

ii

iii

IV

V

vi

vii°

Scale degree
(major mode)

tonic

supertonic

mediant

subdominant

dominant

submediant

leading tone

Roman numeral

i

ii°

III

iv

V

VI

(♭)VII

vii°

Scale degree
(minor mode)

tonic

supertonic

mediant

subdominant

dominant

submediant

subtonic

leading tone

Modern non-English-speaking usage

The above uses are customary for English-speaking countries. Although many of them are also maintained in other countries, those countries have additional uses for Roman numerals which are unknown in English-speaking regions.

The Catalan, French, Hungarian, Italian, Portuguese, Polish, Romanian, Russian and Spanish languages use capital Roman numerals to denote centuries. For example, XVIII refers to the eighteenth century, so as to avoid confusion between the 18th century and the 1800s. (The Italians also take the opposite approach, basing names of centuries on the digits of the years; quattrocento for example is a common Italian name for secolo XV, the fifteenth century.) Some scholars in English-speaking countries have adopted the former method.

In Italy, Poland, Russia, Central Europe, and in Portuguese, Romanian and Serbian languages, mixed Roman and Arabic numerals are used to record dates (usually on tombstones, but also elsewhere, such as in formal letters and official documents). Just as an old clock recorded the hour by Roman numerals while the minutes were measured in Arabic numerals, the month is written in Roman numerals while the day is in Arabic numerals: 14-VI-1789 is 14 June 1789. This is how dates are inscribed on the walls of the Kremlin, for example. This method has the advantage that days and months are not confused in rapid note-taking, and that any range of days or months can be expressed without confusion. For instance, V-VIII is May to August, while 1-V-31-VIII is May 1 to August 31.

In Eastern Europe, especially the Baltic nations, Roman numerals are used to represent the days of the week in hours-of-operation signs displayed in windows or on doors of businesses. Monday is represented by I, which is the initial day of the week. Sunday is represented by VII, which is the final day of the week. The hours of operation signs are tables composed of two columns where the left column is the day of the week in Roman numerals and the right column is a range of hours of operation from starting time to closing time. The following example hours-of-operation table would be for a business whose hours of operation are 9:30AM to 5:30PM on Mondays, Wednesdays, and Thursdays; 9:30AM to 7:00PM on Tuesdays and Fridays; and 9:30AM to 1:00PM on Saturdays; and which is closed on Sundays.

I

9:30–17:30

II

9:30–19:00

III

9:30–17:30

IV

9:30–17:30

V

9:30–19:00

VI

9:30–13:00

VII

—

Since the French use capital Roman numerals to refer to the quarters of the year (III is the third quarter), and this has become the norm in some European standards organisation, the mixed Roman–Arabic method of recording the date has switched to lowercase Roman numerals in many circles, as 4-viii-1961. (ISO has since specified that dates should be given in all Arabic numerals, in ISO 8601 formats.)

In geometry, Roman numerals are often used to show lines of equal length.

In Romania and Serbia to lesser extent, Roman numerals are used for floor numbering. Likewise apartments in central Amsterdam are indicated as 138-III, with both an Arabic numeral (number of the block or house) and a Roman numeral (floor number). The apartment on the ground floor is indicated as '138-huis'.

Roman numerals are rarely used in Asia. The motion picture rating system in Hong Kong uses categories I, IIA, IIB, and III based on Roman numerals.

Alternate forms

Roman Numerals, 16th century

In the Middle Ages, Latin writers used a horizontal line above a particular numeral to represent one thousand times that numeral, and additional vertical lines on both sides of the numeral to denote one hundred times the number, as in these examples:

I for one thousand

V for five thousand

|I| for one hundred thousand

|V| for five hundred thousand

The same overline was also used with a different meaning, to clarify that the characters were numerals. Sometimes both underline and overline were used, e. g. MCMLXVII, and in certain (serif) typefaces, particularly Times New Roman, the capital letters when used without spaces simulates the appearance of the under/over bar, e.g. MCMLXVII.

Sometimes 500, usually D, was written as I followed by an apostrophus (which resembles a backwards C, i.e. Ɔ), while 1,000, usually M, was written as CIƆ. This is believed to be a system of encasing numbers to denote thousands (imagine the Cs as parentheses). This system has its origins from Etruscan numeral usage. The D and M symbols to represent 500 and 1,000 were most likely derived from IƆ and CIƆ, respectively.

An extra Ɔ denoted 500, and multiple extra Ɔs are used to denote 5,000, 50,000, etc. For example:

Base number

CIƆ = 1,000

CCIƆƆ = 10,000

CCCIƆƆƆ = 100,000

1 extra Ɔ

IƆ = 500

CIƆƆ = 1,500

CCIƆƆƆ = 10,500

CCCIƆƆƆƆ = 100,500

2 extra Ɔs

IƆƆ = 5,000

CCIƆƆƆƆ = 15,000

CCCIƆƆƆƆƆ = 105,000

3 extra Ɔs

IƆƆƆ = 50,000

CCCIƆƆƆƆƆƆ = 150,000

Sometimes CIƆ was reduced to an lemniscate symbol (ↀ) for denoting 1,000. John Wallis is often credited for introducing this symbol to represent infinity (∞), and one conjecture is that he based it on this usage, since 1,000 was hyperbolically used to represent very large numbers. Similarly, 5,000 (IƆƆ) was reduced to ↁ; and 10,000 (CCIƆƆ) was reduced to ↂ

In medieval times, before the letter j emerged as a distinct letter, a series of letters i in Roman numerals was commonly ended with a flourish; hence they actually looked like ij, iij, iiij, etc. This proved useful in preventing fraud, as it was impossible, for example, to add another i to vij to get viij. This practice is now merely an antiquarian's note; it is never used.

Medieval Roman numerals

Most uniquely, during the Middle Ages there came about a unique, more comprehensive shorthand for writing Roman numerals, called today the "medieval Roman numerals." This system used almost every other letter of the Roman alphabet to stand as abbreviations for more longhand numbers (usually those that consisted of repetitions of the same symbol). They are still listed today in most dictionaries, although through disfavor are primarily out of use.[4]

Modern
number

Medieval
abbreviation

Notes

5

A

Resembles an upside-down V. Also said to equal 500.

7

S, Z

Presumed abbreviation of septem, Latin for 7.

11

O

Presumed abbreviation of (e.g.) onze, French for 11.

40

F

Presumed abbreviation of English forty.

70

S

Also could stand for 7, and has same etymology.

80

R

90

N

Presumed abbreviation of nonaginta, Latin for 90.

150

Y

Possibly derived from the lowercase y's shape.

151

K

This unusual abbreviation's origin is unknown; it has also been said to stand for 250.

An accurate way to write large numbers in Roman numerals is to handle first the thousands, then hundreds, then tens, then units.
Example: the number 1988.
One thousand is M, nine hundred is CM, eighty is LXXX, eight is VIII.
Put it together: MCMLXXXVIII.

Unicode

Unicode has a number of characters specifically designated as Roman numerals, as part of the Number Forms range from U+2160 to U+2183. For example, MCMLXXXVIII could alternatively be written as ⅯⅭⅯⅬⅩⅩⅩⅧ. This range includes both upper- and lowercase numerals, as well as pre-combined glyphs for numbers up to 12 (Ⅻ or XII), mainly intended for the clock faces for compatibility with large East-Asian character sets such as JIS X 0213 that provide these characters. The pre-combined glyphs should only be used to represent the individual numbers where the use of individual glyphs is not wanted, and not to replace compounded numbers. Additionally, glyphs exist for alternate forms of 1000, 5000, and 10000.

Table of Roman numerals in Unicode

0

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

U+2160

Ⅰ

Ⅱ

Ⅲ

Ⅳ

Ⅴ

Ⅵ

Ⅶ

Ⅷ

Ⅸ

Ⅹ

Ⅺ

Ⅻ

Ⅼ

Ⅽ

Ⅾ

Ⅿ

U+2170

ⅰ

ⅱ

ⅲ

ⅳ

ⅴ

ⅵ

ⅶ

ⅷ

ⅸ

ⅹ

ⅺ

ⅻ

ⅼ

ⅽ

ⅾ

ⅿ

U+2180

ↀ

ↁ

ↂ

Ↄ

ↄ

ↅ

ↆ

ↇ

ↈ

The characters in the range U+2160–217F are present only for compatibility with other character set standards which provide these characters. For ordinary uses, the standard Latin letters are preferred. Displaying these characters requires a program that can handle Unicode and a font that contains appropriate glyphs for them.

Games

After the Renaissance, the Roman system could also be used to write chronograms. It was common to put in the first page of a book some phrase, so that when adding the I, V, X, L, C, D, M present in the phrase, the reader would obtain a number, usually the year of publication. The phrase was often (but not always) in Latin, as chronograms can be rendered in any language that utilises the Roman alphabet.

Mnemonic devices

There are several mnemonics that can be useful in remembering the Roman numeral system.

The following mnemonics recall the order of Roman numeral values above ten, with L being 50, C being 100, D being 500, and M being 1000.

Let Caesar Destroy Mountains

Lucky Cows Drink Milk

Lucy Can't Drink Milk

Lazy Cows Don't Moo

Little Cats Drink Milk

LCDMonitor

A longer mnemonic helps to recall the order of Roman numerals from large to small.